By J. L. Speranza
---- dedicated to R. B. Jones
------- for the Grice Club, etc.
I WAS RECENTLY DISCUSSING some of Grice`s points re: the analysis, or subanalysis, rather, of propositions (or propositional complexes, out of simplexes). Grice relies, explicitly on second-orde set theory:
"[W]e associate with the subject-expression of a canonically
formulated sentence [expressing a proposition, bearer of truth] a set of
at least second order."
--- He considers three cases:
First case: singular entities (as in "France", or "South of France")
"[1] If the subject-expression is a singular name, its ontological
correlate will be the singleton of the singleton of the entity which bears
that name. ..."
Second case: indefinite phrases (like "Somewhere in the South of France").
"[2] If the subject-expression is an indefinite quantificational phrase ..., its ontological correlate will be the set of all singletons whose sole element if an item belonging to the extension of the predicate to which the indefinite modifier is attached." ...
Third case: universal quantification (as in "All of France")
"[3] If the subject-expression is a universal quantificational [all-together, rather than one-at-a-time] phrase, ... its ontological correlate will be the singleton whose sole element is the set which forms the extension of the predicate to which
the universal quantifier is attached."
(Reply to Richards, p. 77ff).
The essay is not easily distributed, since it is in a festschrift, and who buys festscrhifts, but hey.
R. B. Jones may wonder about the technicality here of a second-order set theory.
In any case, it struck me as an intersesting case of Grice relying on some useful formalism (as per the philosophy or theory of mathematics and computer science) to deal with, er, the vernacular.
I suppose second-order set entities are no mystery to most, but it may do to revisit Grice´s motivations in so doing. On a longer day, though!
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Of course, I have to comment on this one.
ReplyDeleteFirst a little terminological trivia.
The first-order/second-order/higher-order terminology is thoroughly confusing because of first order set theory.
One naturally thinks sets as having orders, which correspond to another term, the "rank" of a set.
One thinks of individuals as having order zero, of sets of individuals as having order 1 and sets of sets of individuals as having order 2.
This is the usage which Grice is working with in the matters discussed here.
However, when we move from talking of the order of particular sets, to talking about set theories, there is a completely different terminology (though coming from the same origins).
This difference arises because modern set theory is usually formalised in first order logic, and is called in its entirety "first order set theory" even though the objects in the domain of discourse have every conceivable order, from zero up to orders which noone but professional set theorists can comprehend (objects of these orders are the subject of "large cardinal axioms").
Though most set theorists work with first order set theories, and most mathematicians work in first order set theory (if informally), there is also second set theory. This is what you get when you axiomatise set theory in second order logic. It doesn't actually give you higher orders of sets, instead it allows you to quantify over things that most would call classes, which in a first order set theory don't exist, such as the class of all sets.
As to Grice's correspondences, I don't recall the context for this. One would expect such an account to be a part of a general account of the semantics of some language, and it would then be judged according to whether the whole account fitted together and correctly expressed the semantics of the language in question.
The issues at stake seem all such as could be addressed in a simple type theory such as Church's (essentially the same as the Higher Order Logic supported by ProofPower), which contains (or in which can be defined) definite and indefinite description operators.
Grice's use of second order sets in this context seems a little anomalous to me, though it is plausible that it could be made to work.
Most logics are given a semantics in terms of interpretations, i.e. the truth conditions of sentences are given relative to some interpretation of the language. In HOL such an interpretation would include a value for the indefinite description operator which assigned a single individual to each set to which the operator could be applied.
In such a treatment, the collection of possible values obtained for a description would not appear as the denotation (which is the same kind of thing as Grice's correlates) of the description under any interpretation.
You would get just one of them in each interpretation, and to find all the possibities you would have to look at all possible interpretations.
The effect of this is that you get away with objects of a lower order.
Even if you are being (as Grice here seems to be) rather Aristotelian in having the same kind of correlate for individuals and universals (held by some, e.g. Russell, to be simply a cock-up in traditional logic) then you still only need first order sets as their correlates, a singleton for a particular and possibly larger set for a universal.
I am now myself slightly more familiar with Grice's talk about correlates in his "vacuous names", of which you were so good as to supply me a copy last year, and which I then scrutinised formally with ProofPower. The result was:
rbjones.com/rbjpub/pp/doc/t037.pdf
which I recently revisited for a bit of tidying.
Roger Jones
What an excellent thing! I will re-read Grice's commentary. Especially why he switches from a first-order to a second-order, in his account of 'complexes'.
ReplyDeleteGrice's "Reply to Richards" is apparently partially available online in Google books. "Philosophical Grounds of Rationality" being the title of the book. It is in this reply, of course, that Grice fights the monsters which Jones knows so well! ("Minimalism, Phenomenalism" and the 10 rest of them!).
Thanks again for commentary, and when I have time ("_the_ time" sounds presumptious) to re-type Grice's little adventure in the theory of 'types' (different use of 'type'), will do (i.e. type).
Having read so much of Stanley's 'contextualist' claims, I am feeling force to have to explicate most of my own explicatures, which, needless to say...